Majid Jandaghi Alaee � Greg Ivey

Charitha Pattiaratchi

Secondary circulation induced by flow curvature and Coriolis effectsaround headlands and islands

Received: 2 February 2003 /Accepted: 9 July 2003� Springer-Verlag 2004

Abstract Previous studies have shown that flow curva-ture in river bends generates a secondary circulation inthe plane normal to the mean flow direction. A similarcirculation pattern is shown to exist in oceanic situationswhen flows are subject to curvature, mainly due tointeraction with topographic features. However, it isshown that, due to differences between oceanic condi-tions and river bends, theory and prediction methodsbased on the assumptions for river bends are invalid foroceanic flows. Via scaling arguments based on theequations of motion, that include both the effects of flowcurvature and the Coriolis force, parameters that governthe different flow regimes are identified. The maximumstrength of the secondary flow is derived for each flowregime and is verified using a three-dimensional (3-D)numerical model applied to an idealized island. It is alsoshown that upwelling, due to the generation of second-ary flow, occurs off the tips of the headland or island,and its influence can extend far downstream.

Keywords Secondary circulation � Flow curvature �Coriolis effects � Headlands � Islands

1 Introduction

Studies on the flow structure in river bends have con-firmed that vertically sheared, curved flows producesecondary circulation in the plane normal to the meanflow direction (e.g. Rosovski 1957; Thorne and Hey1979; de Vriend 1981). The curvature-induced secondarycirculation is the result of a local imbalance between thevertically varying centrifugal force and the cross-streampressure gradient (e.g. Kalkwijk and Booij 1986,

henceforth KB). This force imbalance always results inflow towards the inside of the bend in the lower part ofthe water column and an outward flow near the surface.In large-scale flows where rotation is important, anyimbalance between the cross-stream pressure gradientand the Coriolis acceleration can also result in a sec-ondary flow. The orientation of this type of secondarycirculation associated with the Coriolis accelerationdepends on the direction of the flow.

KB proposed an analytical model to investigate thesecondary flow for the case of steady, constant curvatureflows. Dealing with the transverse momentum equationin the curvilinear system, KB suggested a simple balancebetween transverse friction and deviation of the drivingforce (curvature and/or Coriolis) from its mean. Takinga logarithmic profile for the flow velocity over the depth,model predictions of transverse velocity showed goodagreement with experimental data from curved flumeswith shallow rectangular cross-sections.

Garrett and Loucks (1976) suggested that oceaniccurving flows around headlands may also result in thegeneration of secondary circulation. Tidal current datain the vicinity of Gay Head, USA, reported by Geyer(1993), clearly showed the presence of the secondarycirculation. The observed transverse secondary flow of0.13 m s�1 (corresponding to 15–20% of the streamwiseflow) was much stronger than the 0.03 m s�1, computedby the model of KB. This result revealed what could bea significant influence of the secondary flow on the flowdynamics around headlands. Geyer (1993) suggestedthat two factors were most likely responsible for thediscrepancy: (1) enhanced shear due to the differencebetween the actual velocity profile and the logarithmicprofile in the streamwise flow; and, (2) reduced verticalmixing caused by stratification. However, for oscilla-tory oceanic curving flows such as tidal flows aroundheadlands, the time-dependent term and, more impor-tantly, the varying characteristics of the flow in thestreamwise direction, neglected in KB, may have con-tributed to the discrepancy. The observations by Geyer(1993) indicate that the model by KB may not be

Ocean Dynamics (2004) 54: 27–38DOI 10.1007/s10236-003-0058-3

Responsible Editor: Richard Signell

M.J. Alaee Æ G. Ivey Æ C. Pattiaratchi (&)Centre for Water Research,The University of Western Australia,35, Stirling Highway, Crawley 6009, Australiae-mail: pattiara@cwr.uwa.edu.au

applicable to the oceanic conditions such as Gay Headand similar sites.

Garrett and Loucks (1976) suggested that secondaryflow at headlands may be sufficiently strong to induceupwelling, the offshore flow in the upper part of the watercolumn being replaced locally by onshore flow in thebottom boundary layer. So far, only one mechanism hasbeen proposed for upwelling in island or headland wakes:Ekman pumping (Greenspan 1968: Pedlosky 1979),causing upwelling in the centre of island wake eddies(Wolanski et al. 1984, 1994, 1996; Deleersnijder et al.1993). However, there have been observations ofupwelling off headlands and islands in the absence of aneddy (e.g. St John and Pond 1992). Alaee et al. (1998)reported upwelling in the wake of Rottnest Island, wherean eddy was unlikely to be generated in the wake region.Thus, upwelling as a result of the secondary circulationmay be responsible for the observed vertical water mo-tion in wakes in the absence of an eddy.

This paper focuses on the generation of secondarycirculation due to flow curvature with the aim ofdeveloping a prediction method to estimate the strengthof the secondary circulation. Dynamics of secondaryflows and the models utilized in the previous studies arereviewed in Section 2. In this section, two non-dimen-sional numbers are identified and a classification schemein the form of a flow regime diagram is suggested. Thetransverse velocities are predicted for each of the flowregimes using simple scaling arguments. The 3-Dnumerical model (HAMSOM) is described in Section 3and the remainder of the paper presents the results ofnumerical simulations over a wide parameter range inorder to compare the results with prediction methods.Finally, the applicability of the models and the occur-rence of upwelling are discussed in Section 5.

2 The dynamics of secondary circulation

A curvilinear coordinate system s, n and z is considered(Fig. 1) with the streamwise coordinate s oriented in thedirection of vertically averaged flow and the cross-

stream coordinate n oriented normal to the flow. Fol-lowing KB, the momentum equation in the n directioncan be written as

@un

@tþ @usun

@sþ @u2n@nþ @uzun

@zþ 2

usun

Rn

þ u2n � u2

s

Rsþ fus þ

1

q@p@nþ friction terms ¼ 0 ; ð1Þ

where Rs and Rn are the radii of curvature of thestreamlines of the depth-averaged flow in the streamwiseand normal directions, respectively, p is the pressure, qthe mass density and f the Coriolis parameter. Assumingun � us, uniform density, neglecting friction in verticalplanes and vertical advection Eq. (1) can be simplified,using the hydrostatic pressure assumption and frictionterms parameterized in terms of an eddy viscosity:

@un

@tþ us

@un

@s� u2

s

Rsþ g

@g@nþ fus �

@

@zKz@un

@z

� �¼ 0 : ð2Þ

Here, g is the water level, g is the acceleration due togravity and Kz the vertical eddy viscosity. By definition,the depth-averaged transverse velocity, un, is zero. KBaveraged Eq. (2) over depth and obtained:

us@un

@s� u2s

Rsþ g

@g@nþ f us þ

sn

qh¼ 0 ; ð3Þ

where sn is the bottom friction in the n direction and h isthe water depth. Eliminating the free surface slope termby combining Eqs. (2) and (3), KB found an expressionfor the secondary flow as:

@un

@tþ us

@un

@s� us

@un

@s� @

@zKz@un

@z

� �� sn

qh

¼ � u2s � u2sRs

� f ðus � usÞ : ð4Þ

The right-hand side of Eq. (4) identifies two distinctmechanisms leading to transverse motion: flow curva-ture and the Coriolis effects due to the deviation of usfrom the vertical mean, in both cases.

To obtain an analytical solution for steady flows incurved open channels, KB ignored all streamwise vari-ations and assumed a logarithmic profile for us as well asa parabolic form for the vertical eddy viscosity. Underthese assumptions Eq. (4) simplifies to:

� @

@zKz@un

@z

� �� sn

qh¼ driving force ; ð5Þ

which was solved analytically by KB. The results were ingood agreement with the laboratory measurements ofconstant curvature channels with uniform rectangularcross-sections. It was proposed that beyond an initialrelaxation length in a circular channel bend, the sec-ondary circulation reaches its fully developed state cor-responding to the maximum transverse velocity, un;max.From the KB model, under the assumption of logarith-mic profiles in the vertical, the transverse velocity in-duced by either flow curvature (unb;max) or the CoriolisFig. 1 The curvilinear coordinate system

28

acceleration (unc;max) in the surface layer can be estimated(for a given drag coefficient CD ¼ 3� 10�3) as follows:

unb;max � kbðsÞ6jusjh

Rsð6aÞ

unc;max � kcðsÞ3fh ; ð6bÞwhere kbðsÞ and kcðsÞ are functions of the streamwisecoordinate and vary between 0 and 1. In fact, they arethe ratio of local un to the fully developed un. Based onthese assumptions, Geyer (1993) estimated the maxi-mum strength of un for tidal flow at Gay Head andfound that the observed secondary flow was consistentlyup to four times stronger than the predicted values fromEq. (6). This indicates that the omission or retention ofsome of the terms on the LHS of Eq. (4) may not bejustified in many field situations. Comparison betweenthe terms on the LHS of Eq. (4) using data from GayHead indicates that the non-linear terms play a key rolewhile viscous forces are negligible compared to bottomfriction. Neglecting the viscous term and the verticallyaveraged non-linear term in comparison to other terms,Eq. (4) can be rewritten in the steady state as:

us@un

@s� sn

qh¼ � u2

s � u2s

Rs� f ðus � usÞ : ð7Þ

The relative importance of advection to friction in theLHS of Eq. (7) is given by:

us@un

@s=

sn

qh� h

LCD¼ Ref ; ð8Þ

where L is the streamwise length scale and Ref has beendefined as the equivalent Reynolds number (e.g. Pingreeand Maddock 1980; Pattiaratchi et al. 1987; Signell andGeyer 1991). In a similar manner, the relative importanceof the twopotential driving forceson theRHSofEq. (7) is:

u2s � u2s

Rs=f ðus � usÞ �

ab

� �us

fRs� Rom : ð9Þ

Here, we have introduced a and b as characteristicparameters of the streamwise velocity defined by:

a ¼ 1� u2su2s

ð10Þ

b ¼ 1� us

usð11Þ

and Rom as the modified Rossby number. Although it isexplained later that a and b may be approximated bytwo constants so that Rom is equivalent to the Rossbynumber (Ro ¼ 2us=fRs), we keep Rom in order to comparethe numerical results with the predictions. It means that

for the comparison, we consider the effects of the localprofile of the streamwise velocity in the prediction of un,while in practice these effects will be included in a con-stant coefficient (see below).

Rom and Ref represent the relative magnitudes of thegeneration and dissipation of the momentum in thesecondary flow. On this basis, flow regimes in relation tosecondary flow may be categorized by considering valuesof Rom and Ref . For either of these non-dimensionalnumbers there would be a value indicating an approxi-mate balance between the forcing mechanisms. There-fore, four flow regimes may be defined as shown inTable 1 (see also Fig. 2). Now, for each of the flowregimes the strength of the secondary circulation, un, canbe predicted considering a balance between the remain-ing forces in Eq. (7). The strength of the secondary flow(un) associated with each of the flow regimes A, B, C andD (Fig. 2) can be predicted from the primary forcebalance as follows:

Regime A: un �fbhCD

ð13Þ

Regime B: un � fbL ð14Þ

Regime C: un �ahUs

CDRsð15Þ

Regime D: un �abUs

Rs: ð16Þ

We now examine these predictions of un (Table 1) usinga three-dimensional numerical model to simulate theconditions for the different flow regimes. The corre-sponding results, comparison and discussion are pre-sented in the subsequent sections.

Fig. 2 The flow regime diagram based upon the two non-dimensionalnumbers, Rom ¼ ða=bÞðbU=RsÞ and Ref ¼ h=bCD

Table 1 Flow regimes and thecorresponding dominantforce balances

Regime A Balance between bottom friction and Coriolis forces Rom < 1 and Ref < 1Regime B Balance between inertia and Coriolis forces Rom < 1 and Ref > 1Regime C Balance between bottom friction and centrifugal forces Rom > 1 and Ref < 1Regime D Balance between inertia and centrifugal forces Rom > 1 and Ref > 1

29

3 The numerical model

The three-dimensional primitive equation model,HAMSOM (HAMburg Shelf Ocean Model), wasdeveloped by Backhaus (1985) for the North Sea(Backhaus and Hainbucher 1986). The model is basedupon a semi-implicit numerical scheme described byStronach et al. (1993). However, for the barotropicmode, the model is implicit so that relatively largecomputational time steps can be employed. The HAM-SOM model has been widely adapted to the GermanBight; Juan de Fuca system (Backhaus et al. 1987;Stronach et al. 1993); the South China Sea (Pohlmann1986), Atlantic coast off Spain and Baja California; andSouthwestern Australia (Pattiaratchi et al. 1996).

The model uses fixed permeable horizontal interfacesbetween layers and the equations of continuity andmomentum are vertically integrated over each modellayer. The distribution of pressure is assumed to behydrostatic and the Boussinesq approximation is in-voked. The model uses a finite difference scheme withthe variables distributed over the grid according to theArakawa C grid (Arakawa and Lamb 1977). The solu-tion technique is described in Backhaus (1985) andStronach et al. (1993).

HAMSOM includes the choice of either one of threeindependent turbulent closure schemes for the computa-tion of the vertical eddy viscosity (Kz). These threeschemes are based on the Richardson principle (Stronachet al. 1993), the Kochergin closure scheme (Kochergin1987) and constant Kz in the water column. The hori-zontal eddy viscosity (Kx) is considered constant every-where. At the surface and bottom, boundary conditionsare applied which include surface wind stress and bottomshear stress. The former is specified using a drag rela-tionship dependent on the wind speed and direction. Thebottom boundary condition is of the form of the qua-dratic bottom stress. At lateral solid boundaries, the usualno-flux condition is used while the roughness of a solidboundary can be considered by choosing no-slip, half-slipor full-slip boundary conditions. The sea-surface eleva-tion or the flow must be prescribed at open boundaries.

4 The idealized domain

An idealized domain was used to isolate the influence ofthe important terms in Eq. (7) such as friction, centrif-ugal force and the Coriolis force on the development ofthe secondary flow. The model domain (Fig. 3a) con-sisted of a constant depth (h) channel with a 180� 160grid (283 � 185 km2). An elliptic island with slopingsides was specified at the channel centre (Fig. 3b). Theisland was defined by:

y ¼ b 2� x� x0a

� �2� �12

þy0 ; ð17Þ

where a and b denote the major and minor axes of theellipse at the surface and x0, y0 are the coordinates at thecentre of the channel.

The ability of the HAMSOM model to handle avariable grid was used to refine resolution around theisland, where the secondary flow is likely to be strong.Hence, utilizing four different grid sizes (250, 1000,5000 and 15 000 m) in the domain ensured that theboundary effects did not influence the area of interest(Fig. 3a). It was also possible to choose vertical layerswith different thickness. Thus, in all experiments fivelayers (5, 10, 10, 10 and 5 m thick, from the surface tothe bottom, respectively) were considered. These runswere conducted in barotropic mode (see Stronach et al.1993).

The model included two open boundary conditions,the northern and southern boundaries as seen inFig. 3a. In the upstream open boundary an inflow,uniform along both the width and depth of thechannel, was introduced while the model was allowedto compute the outflow characteristics using the Or-lanski radiation condition (Orlanski 1976). A no-slipboundary condition was used for all the lateralsolid boundaries. Each experimental run was startedfrom rest and continued until the flow was in a steadystate.

5 The numerical experiments

For each of the flow regimes defined in Section 2, severalmodel runs were conducted. In all the runs, the idealizedgeometry was used with h ¼ 16 or 40 m and usinga ¼ 20 km, b ¼ 10 km, S ¼ 1 km (Fig. 3b); The forcingwas a uniform inflow of either 3.2 or 8 m2 s�1

(m�1width) equivalent to a uniform flow velocity of0.20 m s�1 which resulted in current speeds off tips of theidealized island similar to the typical current speedsaround Rottnest Island (see Alaee et al. 1998) whichprovided the motivation for the present work. To avoidinstabilities due to surface waves, the inflow wasincreased gradually from zero to its final value andremained constant afterwards. In all of the model runs,the results indicated an attached flow condition, wherethe flow separation either does not occur or does notresult in eddy generation in the wake region (see Figs. 4aand 5a).

As explained below, the interaction between theeffects of Coriolis and flow curvature leads to two dif-ferent flow structures around the tips of the island. Inorder to assess the prediction method for the strength ofthe secondary flow, 38 sampling locations were selectedaround each tip of the island, where the presence ofstrong secondary circulation was expected (Fig. 3b). Atthese locations the predicted transverse velocityunðpredictedÞ or unp; was calculated based upon the scalingestimates given in Eqs. (13) to (16), and compared tounðmodelÞ, or unm; obtained from the numerical modellingresults, for each of the different regimes.

30

5.1 Regime A (Rom < 1 and Ref < 1)

Five numerical experiments were conducted corre-sponding to regime A, as described in Table 2. To satisfythe conditions, the vertical eddy viscosity and the dragcoefficient for bottom friction were set to 0.005 m2 s�1

and 0.01, respectively. Various Coriolis parameters, f(for the Southern Hemisphere), were chosen for theseexperiments to produce Rom of the order of 0.1. SinceRom is dependent on the local velocity profile and localradius of flow curvature, we computed Rom based onEq. (9) for all the sampling locations. It was found thatRom was O(0.1) for locations away from the island.

For each numerical run, the distribution of thetransverse velocity in the surface layer was obtained. Forexample, Fig. 4a and b show the velocity field and thedistribution of the unm in the surface layer, for the runusing f ¼ 9:6� 10�4 s�1 (run Infcorbf26). To predict thetransverse velocity, unp, from Eq. (13), b was computedfrom the local vertical profile of the streamwise velocity,

obtained from the numerical results, at each of thesampling locations. The results indicated that b variedover a range of 0.15 to 0.2, for various locations and inbetween different runs. The predicted transverse velocityunp was compared to unm; measured directly from thenumerical results. This comparison implied that for allthe locations around the island, the flow curvature stillinfluences the ensuing secondary circulation. With dis-tance away from the island, the streamlines tend to bestraight, Rs !1 (similar to cases with b!1) and thusRef ! 0 and Rom ! 0. The comparison between thepredicted and the measured transverse velocity for thesefar-offshore locations showed good agreement betweenunp and unm (Fig. 6).

5.2 Regime B (Rom < 1 and Ref > 1)

Ref > 1 indicates that the non-linear term is the domi-nant term (Table 1) in the LHS of Eq. (7), implying

Fig. 3a Plan view of theidealized domain indicating thedifferent grid sizes used in thenumerical simulations. Auniform depth of h was adoptedfor the whole domain. b The areawith the finest grid size (250 m)and the elliptical island with thegreater and less diameter of aand b. The sloping side of theisland reaches the depth of H(dashed line) in a distance ofS. The sampling locations areshown in black and white dots

31

@un=@s 6¼ 0. Since f is constant over the domain, thevariation of un in the streamwise direction results fromvariations in flow-curvature effects. Thus, the sampledlocations must be in the region of the flow curvature, butfrom Section 1, the locations may not be very close tothe island either, because Rom increases as Rs decreases inthe immediate vicinity of the island boundaries.

Four numerical runs were conducted with parameterssimilar to those used in the regime A except that thebottom drag coefficient was set to 0.001 to ensure

Ref > 1. To estimate the transverse velocity fromEq. (14), b was computed for all the selected locations.Figure 7 shows the comparison between unm and unp forlocations corresponding to the four numerical runs. Thiscomparison indicates that the predicted transversevelocities are in good agreement with that from thenumerical results.

5.3 Regime C (Rom > 1 and Ref < 1)

Two numerical experiments, with differing waterdepths and horizontal eddy viscosities, were conductedcorresponding to the flow regime C (Table 1). In all theruns, the Coriolis parameter was set to zero to ensurethat the centrifugal force was the only driving force inthe RHS of Eq. (7). To predict the corresponding unpfrom Eq. (15), a was computed at the sampling loca-tions from the local profile of us using the numericalresults. Since unp for the regimes C and D is inverselyproportional to the radius of flow curvature, Rs wascalculated based on the orientation of velocity vectorsin two successive numerical grids, at each samplinglocation (see Kalkwijk and Booij 1986; Chant andWilson 1997) and the results show that the predictedtransverse velocity unp is in good agreement with unm(Fig. 8). However, for a few locations adjacent to theisland or far offshore, the predicted transverse velocitiesdo not correspond with the numerical results. Thisdiscrepancy is likely to be due to inaccuracies in thedetermination of Rs.

5.4 Regime D (Rom > 1 and Ref > 1)

Several numerical runs with various parameter valueswere conducted (Table 2). Similar to the preceding sec-tion, the Coriolis term was neglected (f ¼ 0) andtherefore the flow curvature term was the only drivingforce in Eq. (7). In order to compare the predictedtransverse velocity from Eq. (16) to the numericalresults, a and Rs were computed for each samplinglocation. Figure 9 shows that the predicted transversevelocity from Eq. (16) has reasonable agreement withunm obtained from the numerical results, despite inac-curacy in the estimation of Rs for the neighbouringlocations to the island or far offshore sampling points.

6 Discussion

The flow classification described in Table 1 can beshown as a flow regime diagram (Fig. 1). The verticaldiffusion coefficient influences the transverse velocitythrough the streamwise velocity profiles and hence thecoefficients a and b.

For the four regimes in the flow-regime diagram(Fig. 1), the maximum transverse velocity can beestimated, using Eqs. (13) to (16), as:

Fig. 4a The velocity field in the surface layer from the numericalmodelling results of the case corresponding to the flow regime A withh ¼ 16 m, Kz ¼ 0:005 m2 s�1, Kx ¼ 1 m2 s�1, CD ¼ 0:01 andf ¼ 9:6� 10�5 s�1 (run Infcorbf26). b Distribution of the transversevelocity in the surface layer for run Infcorbf26. Positive and negativevelocities indicate the secondary circulation flowing in the positive andnegative n direction, respectively

32

Un ¼ KAfhCD

ðfor Rom < 1 and Ref < 1Þ ð17Þ

Un ¼ KBfb ðfor Rom < 1 and Ref > 1Þ ð18Þ

Un ¼ KChUs

CDRsðfor Rom > 1 and Ref < 1Þ ð19Þ

Un ¼ KDbUs

Rsðfor Rom > 1 and Ref > 1Þ ; ð20Þ

where Un represents the maximum transverse velocitynear the surface, the streamwise length scale L is taken asthe island ellipse scale b, and the coefficients KA, KB, KCand KD the profile factors a and b. Flow with a specificvertical profile and a given bottom drag coefficientresults in constant values for the profile factors. Forinstance, assuming a logarithmic profile (see Kalkwijkand Booij 1986), CD ¼ 3� 10�3 yields a ¼ 0:22 andb ¼ 0:12 and hence a=b ¼ 1:8. Note that the approachhere has been to avoid the a priori assumption that allstreamwise velocities are logarithmic over depth and touse the simple but plausible values of bottom drag andvertical viscosity in our model runs with water depths of16 and 40 m. In this respect, the numerical experimentsindicated as a and b do not vary widely and the ratio a=bmay be approximated by 2. Therefore the modifiedRossby number, defined in Eq. (9), is simplified toRom ¼ 2Us=fRs.

6.1 Flow regimes A and B

In the flow regimes A and B, the influence of flow cur-vature in the generation of the secondary circulation isnegligible compared to Coriolis effects. Figure 10ademonstrates the comparison between unp and unm,corresponding to the flow regime A. As explained inSection 4.1, the locations matching the regime A arelocated far offshore from the island to be free from theeffects of the flow curvature. Since the vertical structureof the flow is uniform in the far-offshore region, all thelocations possess identical streamwise flow structure aswell as transverse circulation. On this basis each datapoint in Fig. 10a represents the whole offshore regionfrom the results of one numerical experiment. The bestfit through the data indicatesKA ¼ 0:026 for in Eq. (17)and taking CD ¼ 0:003, similar to the case of Gay Head(Geyer 1993), Eq. (17) is simplified to:

Un � 9fh : ð21ÞEquation (21) is similar to Eq. (6b) from KB for the fullydeveloped secondary circulation except that the constant

Fig. 5a Velocity vectors in the surface layer from the numericalresults of the run InfB2 with h ¼ 40 m, Kz ¼ 0:005 m 2 s�1,Kx ¼ 1 m2 s�1, CD ¼ 0:001. b The transverse velocities in the surfacelayer corresponding to flow structure in InfB2. Positive and negativevelocities indicate the secondary circulation flowing in the positive andnegative n direction, respectively. c The transverse velocities in thebottom layer corresponding to the run InfB2

b

33

value inEq. (21) is three times larger than that in Eq. (6b).For the case of the observations aroundGayHead (Geyer1993), Eq. (6b) yields a value of 0.006 andEq. (21) a valueof 0.018 m s�1, with the latter prediction being muchcloser to the observed Un of about 0.13 m s�1.

Figure 10b illustrates the comparison between unpfrom Eq. (18) and unm measured from the numericalresults at the locations with the regime-B conditions andfrom the least-squares line for the data we obtain a valueof KB ¼ 0:02.

In the runs associated with regime B, variation of thetransverse velocity unm near the island, at whichthe curvature effects become important, is interesting.The transition between curvature- and Coriolis-inducedsecondary circulation can be seen in Fig. 4b: close to theleft tip of the island the transverse circulation induced byflow curvature is stronger than that induced by theCoriolis effects due to the small radius of the curvaturenear the tip and high local flow speed. However, awayfrom the island tip the curvature influences weakenconsiderably due to the very large Rs and thus theCoriolis-induced secondary circulation dominates. Thetransition between these two driving forces is also clear

Table 2 Description of exam-ples of the numerical experi-ments that were conducted inorder to examine the predictionmethod for the strength of thesecondary circulation

Regime Run h (m) CD kx (m2 s)1) Inflow (m

2 s)1) f � 105 (s)1)

A Infcorbf22 16 10)2 1 3.2 8Infcorbf23 16 10)2 1 3.2 11.5Infcorbf25 16 10)2 1 3.2 14Infcorbf26 16 10)2 1 3.2 9.6Infcorbf2 40 10)2 1 8 8

B Infcorb22 16 10)3 1 3.2 8Infcorb23 16 10)3 1 3.2 11.5Infcorb25 16 10)3 1 3.2 14Infcorb26 16 10)3 1 3.2 9.6

C Infbf22 16 10)3 1 3.2 –Infbf2)2 40 10)2 1 8 –

D InfB2 40 10)3 10 8 –Infbf1 40 0.00 1 8 –Infbf2 40 0.003 1 8 –Infbfkx1 40 0.00 1 8 –Infkx1 40 10)3 10 8 –Infkx2 40 10)3 10)1 8 –

Fig. 6 Comparison between the transverse velocity from the predic-tion method and numerical results (for the five runs described in thetext), corresponding to the regime A

Fig. 7 As Fig. 6 for the regime B

Fig. 8 As Fig. 6 for the regime C

34

in the vicinity of the right tip of the island, where theCoriolis- and curvature-induced secondary circulationare of opposite sign. It can be seen that close to the righttip, the curvature effects overcomes the Coriolis effectscausing an offshore unm of 0.01 cm s�1, while at a dis-tance of about 2 km from the tip, an onshore unm of 0.02cm s�1 is generated. This demonstrates that an asym-metry can occur in the distribution of transverse velocityaround an island due to the relative magnitudes of theCoriolis and flow curvature effects.

6.2 Flow regimes C and D

In flow regimes C and D, the centrifugal force due toflow curvature is the dominant forcing in the RHS of

Eq. (7). This forcing is relatively strong off the tip of theisland due to the occurrence of the high flow speed andsmall radius of curvature off the tip. This can be clearlyseen in Fig. 5b, which shows the distribution of thecurvature-induced unm in the surface layer: the strongesttransverse velocities are generated off the tip of theisland, which weaken offshore and downstream.

Figure 11 show the results of the comparison of unpfrom Eqs. (19) and (20) for the flow regimes C and D,respectively, with the corresponding numerical results ofunm. In both cases, spurious data most likely due to theunrealistic computed radius of curvature were not con-sidered. From Fig. 11, two values of 0.11 and 0.27 areobtained for KC and KD, respectively.

Using, CD ¼ 0:003, Eq. (19) can be simplified to:

Un ¼ 35hUs

Rs: ð22Þ

Equation (22) can be compared to Eq. (6a) which wassuggested for the estimate of the maximum transversevelocity induced by flow curvature (Geyer 1993). Equa-tion (6a) predicts Un ¼ 0:028 m s�1, whilst Eq. (22)yieldsUn ¼ 0:166, which compares well with the observedUn � 0:13 m s�1 around Gay Head. According to theflow regime diagram, however, we suggest that the case ofGay Head coincides with regime D as discussed below.

6.3 Application

To use the regime diagram to estimate the strength ofthe secondary circulation in a field situation, it isessential to determine Ref ¼ h=CDb and Rom ¼ 2Us=fRsfor the flow. Then, the maximum transverse velocity inthe surface layer can be estimated using the corre-sponding equation selected from Eqs. (17) to (20).

Consider the case of secondary flow around GayHead, one where the field data is available (Geyer 1993)and where the observed secondary circulation was

Fig. 9 As Fig. 6 for the regime D

Fig. 10a, b The comparison between the transverse velocities resultedfrom the numerical simulations and the predictions for the flowregimes A (a) and B (b) in order to determine the correspondingconstant factors

Fig. 11a, b As Fig. 10 but for the flow regimes C and D

35

forced by tidal flow around a headland. Geyer (1993)used the results of the KB analytical model and showedthat the streamwise flow could be approximated toquasi-steady state. For values of Us ¼ 0:7 m s�1,h ¼ 20 m, b ¼ 2000 m, Rs ¼ 3000 m, f ¼ 1� 10�4 andCD ¼ 0:003, we obtain Ref ¼ 3 and Rom ¼ 5, whichsuggest flow regime D. Using Eq. (20) with KD ¼ 0:27,we obtain Un ¼ 0:13 m s�1, essentially the same as theobservations (� 0:13 m s�1).

As another example, consider the strength of thesecondary circulation around Rottnest Island off theWest Australian coast. During the summer wake ofRottnest Island, a northerly quasi-steady wind-drivencurrent prevails in the Rottnest Island region(Pattiaratchi et al. 1996; Alaee et al. 1998). Field data(Alaee et al. 1998) indicate that wind stress in thetransverse direction is negligible. For values ofUs ¼ 0:35 m s�1, h ¼ 50 m, b ¼ 4500 m, Rs ¼ 3000 m,f ¼ 7:7� 10�5 (Alaee et al. 1998) and CD ¼ 0:003, weobtain Ref ¼ 3:7 and Rom ¼ 3, which coincides with theflow regime D. Using Eq. (20), a maximum transversevelocity of Un ¼ 0:14 m s�1 is predicted for the summerin the vicinity of Rottnest Island.

6.4 Upwelling and downwelling

Garrett and Loucks (1976) suggested that the secondaryflow around headlands may result in localized topo-graphic-induced upwelling. In the plane of the secondarycirculation normal to the main flow, the onshore flownear the bottom is drawn upwards and replaces the off-shore transverse flow near the surface (Fig. 12a). Con-servation of mass equation may be written as:

@us

@sþ @un

@nþ W

h¼ 0 ; ð23Þ

where W is the vertical velocity and h is the total depth.Numerical results presented earlier show that thestreamwise velocity changes slightly over short distancesin the s direction and may neglected in the balance inEq. (23) and the two-dimensional continuity equation isobtained in the plane of the secondary flow.

@un

@n¼ �W

h: ð24Þ

Figure 13 shows the distribution of the verticalvelocity in layer 4 (30 m below the surface) corre-sponding to the numerical results of run InfB2 shown inFig. 5 and reveals that upwelling driven by the offshore-directed transverse velocity is strongest off the tip of theisland and extends a considerable distance downstream.Using this numerical simulation, and extracting datanear the island tip, we obtain DUn ¼ 0:05 m s�1,Dn ¼ 750 m (three numerical grids) and with h ¼ 50 m,we obtain W � 0:003 m s�1 which compares favourablywith the range values in Fig. 13.

Topographic-induced downwelling, in the upstreamside of islands, has been reported by several researchers(e.g. Doty and Oguri 1956; Hamner and Haury 1981).

Fig. 12a, b Schematic diagram indicating the interaction between thesecondary circulation and lateral solid boundary causing upwelling (a)and downwelling (b)

Fig. 13 Distribution of vertical velocity (in m s�1) in the layer 4obtained from the numerical results of InfB2. For grid resolution seetext

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Figure 13 clearly shows a region of downwelling in theupstream side of the idealized island. Related to thisdownwelling, Fig. 5b and c reveals that in the upstreamof the island the transverse velocity is directed onshorenear the surface and offshore near the bottom and thedownwelling is a result of downslope motion on the sideslope of the island, as shown in Fig. 11b. The numericalresults indicate that the downwelling is about 1 order ofmagnitude weaker than upwelling off the tip of the island.

7 Summary and conclusions

In this study, using the numerical simulation of flowstructure around an idealized island or headland, weinvestigated the development of the secondary circula-tion induced by both the flow curvature and the Corioliseffect, for quasi-steady oceanic flows. Simple scaling ofthe transverse momentum equation, in the curvilinearcoordinate system, led us to a flow regime diagram usingtwo non-dimensional numbers, a Reynolds numberRef ¼ h=CDb and a Rossby number Rom ¼ 2Us=fRs. Apredictive method was proposed for the strength of thesecondary flow Un for each flow regime and it was shownthat the predictions were in good agreement with thenumerical results. Nevertheless, a broader range ofsimulations will result in more accurate constant factorsused in the predictive formula.

From the numerical results, the three-dimensionalstructure of the flow implied that the interaction betweenthe secondary circulation and lateral solid boundariesresults in either upwelling or downwelling. It was con-cluded that the strong secondary flow due to flow cur-vature around the island (or headland) tips causes strongupwelling in this vicinity (up to 250 m day�1 from ourresults), which can extend far downstream. Downwel-ling, based on the same mechanism but with reversecurvature, was found along the upstream side of theisland and is highly localized. The results show thatdownwelling is much weaker than upwelling near the tipof the island (or headland). The occurrence of verticalwater motion in the simulations, with no eddy in thewake of the island or headland, implies that an island orheadland may induce upwelling or downwelling evenunder the attached flow condition.

Acknowledgements The first author wishes to thank the Ministry ofCulture and Higher Education and Amir Kabir University ofTechnology for providing a scholarship. The work was undertakenwhilst the first author was a student at the Centre for WaterResearch, The University of Western Australia. This is Centre forWater Research reference ED1269MA.

Notation

a; b semimajor and semiminor axes idealized ellipticalisland

f Coriolis parameter

g acceleration due to gravityh total water depthkb; kc function of skx horizontal eddy viscosity coefficientkz vertical eddy viscosity coefficientn coordinate perpendicular to streamlinep pressures streamwise coordinatet timeun velocity in n direction (transverse velocity)un;b transverse velocity due to curvature effectun;c transverse velocity due to Coriolis effectunm transverse velocity from the numerical resultsunp transverse velocity from the prediction methodx; y rectangular orthogonal coordinatesx0; y0 coordinates of centre of elliptical islandz vertical coordinateCD bottom drag coefficientL streamwise length scaleRn radius of curvature in n directionRs radius of curvature in s directionRef equivalent Reynolds number (¼ h=CDL)Rom modified Rossby number (¼ a=b)(Us=fRs)Un transverse velocity near the surfaceUs streamwise velocity near the surfaceW vertical velocitya shape factor of streamwise velocity profile

(¼ 1� u2s=u2s )

b shape factor of streamwise velocity profile(¼ 1� us=us)

g water levelsn bottom shear stress in n direction

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